High harmonic generation (HHG) is a technique for the production of coherent extreme ultraviolet (XUV, 120-10 nm) and soft x-ray (SXR, 10-0.1 nm) radiation. For this application, high harmonic generation is considered to relate to the generation of harmonics having free space wavelengths of up to 120 nm. The simple theoretical process for HHG was described by Corkum in three steps: (i) ionization of a ground state atom by high intensity, coherent driving radiation via quantum tunnelling of the electron to the continuum; (ii) propagation of the electron in the continuum under the electromagnetic force of the driving radiation; and (iii) recombination with the ion to release a harmonic photon.
It is known to focus linearly polarized incident light from a pulsed laser into a noble gas (e.g. argon) with a peak intensity of order 1014 W cm−2 or greater, generating coherent beams of high harmonic radiation with frequencies that are odd integer harmonics of the frequency of the incident light from the laser.
A significant drawback with HHG is that it is inefficient, due to phase-mismatch, which is caused by the different phase velocities of the fundamental driving radiation and the high harmonic radiation generated. The consequence of the different phase velocities is that harmonically generated photons produced at a particular location within the generation medium will not, in general, be in phase with the harmonically generated photons that were produced from the fundamental radiation field at a different location. This leads to alternating regions of constructive and destructive interference of the high harmonic radiation within the generating medium (e.g. along an optical waveguide), with the intensity of a high harmonic radiation oscillating back to zero with a period of twice the coherence length (Lc), where the coherence length is defined as the distance over which a π phase difference accumulates between the harmonic radiation and the local phase of harmonic generation from the fundamental radiation.
Without implementing additional techniques, the conversion efficiency for generating photons with energies up to about 100 eV is of the order 10−7, and this decreases to as little as 10−15 for harmonic photon energies near 1 keV. Although phase-matching techniques are known, and can significantly improve generation efficiency, such techniques are restricted to relatively low-order harmonics.
Quasi phase-matching (QPM) seeks to enhance harmonic generation efficiency by periodically suppressing harmonic generation with a period of twice the coherence length to eliminate regions of destructive interference and leave only regions in which the generated harmonics interference constructively with the harmonic radiation generated elsewhere in the generation medium.
A first approach to producing linearly polarized high harmonic radiation by QPM has been to provide longitudinal, periodic modulation of the non-linear properties along the optical propagation axis. In one example, it is known to provide alternating regions of argon and nitrogen with a period of twice the coherence length, with constructive interference occurring within the argon regions, and with destructive interference being suppressed in the nitrogen regions. In a further example, it is known to periodically modulate the diameter of the hollow core of a gas-filled hollow-core optical waveguide, with the modulation period matched to the coherence length, so that the intensity of the fundamental wave radiation propagating through the hollow core is modulated periodically along the propagation axis of the optical waveguide, and hence to modulate the rate of generation of the high harmonic periodically. However, this approach is limited to relatively low-order harmonics since high-order harmonics have short coherence lengths, whereas modulation of the fundamental wave (i.e. the driving beam) occurs over a distance of order the Rayleigh range (e.g. up to a few millimeters); hence, as the period of the modulation is reduced, the modulation of the intensity of the fundamental wave is also reduced.
A second approach to producing linearly polarized high harmonic radiation by QPM has been “intensity beating” of two different modes of a conventional hollow core waveguide. In this approach, a single linearly polarized input beam induces two different waveguide modes of the waveguide, which are both linearly polarized with parallel polarizations. The excited modes beat, as they propagate along the propagation axis of the waveguide, causing the axial intensity of the propagating radiation to be modulated with a period determined by the different wave-vectors of the modes. The key requirement for controlled beating of waveguide modes is controlled mode excitation at the entrance of the waveguide, but does not require any special type of waveguide. However, quasi-phase-matching to produce high-order harmonics can be difficult with this technique, since it requires controlled excitation of different waveguide modes, by spatial light modulation techniques.
A third approach to producing linearly polarized high harmonic radiation by QPM has been to interfere radiation of a fundamental wave driving pulse with a weak counter-propagating pulse train of the same frequency, leading to harmonic generation only at points in the harmonic generation medium where the driving pulse does not overlap with one of the pulses in the counter-propagating pulse train. This method suffers from the disadvantage that additional optical systems, and increased driving laser energy, are required to generate the pulse train.
An approach to generate circularly polarized high harmonics is known, in which a polarization gating technique is used with circularly polarized pump radiation incident onto an arrangement of bowtie-type nanoantennae on a silicon substrate, in which alternating nanoantennae are perpendicular. However, disadvantageously, the generation efficiency of this technique is low, and requires precise micro-machining of the nanoantennae. Further, the power of the incident driving radiation may be limited due to a low damage threshold of such structures.